Test Questions for Algebraic Indicators of No or Infinite Solutions

📚 Quick Study Guide

• 🔢 Linear Equations: A linear equation in one variable can have one solution, no solution, or infinitely many solutions.
• 🚫 No Solution: This occurs when the variables cancel out, and you are left with a false statement (e.g., $0 = 5$). This means there's no value for the variable that makes the equation true.
• ♾️ Infinite Solutions: This happens when the variables cancel out, and you're left with a true statement (e.g., $0 = 0$). This means any value for the variable will satisfy the equation.
• ⚖️ Systems of Equations: For a system of two linear equations:
  • If the lines have the same slope and different y-intercepts, there is no solution.
  • If the lines have the same slope and the same y-intercept, there are infinite solutions.
• 📝 Key Idea: Examine the coefficients and constants after simplifying the equation. If the coefficients of the variable are the same on both sides, but the constants are different, there’s no solution. If both the coefficients and constants are the same, there are infinite solutions.

🧪 Practice Quiz

1. What condition indicates that the linear equation $ax + b = cx + d$ has no solution?
   1. $a = c$ and $b = d$
   2. $a \neq c$
   3. $a = c$ and $b \neq d$
   4. $a \neq c$ and $b = d$
2. Which of the following equations has infinitely many solutions?
   1. $3x + 5 = 3x + 5$
   2. $2x - 1 = 2x + 1$
   3. $4x + 2 = 4x - 2$
   4. $x + 3 = x - 3$
3. For what value of $k$ will the equation $2x + 3 = 2x + k$ have no solution?
   1. $k = 3$
   2. $k \neq 3$
   3. $k = 0$
   4. $k = -3$
4. The equation $5(x + 2) = 5x + 10$ has:
   1. One solution
   2. No solution
   3. Infinite solutions
   4. Two solutions
5. Which equation has no solution?
   1. $7x - 2 = 7x - 2$
   2. $6x + 4 = 6x + 1$
   3. $8x + 3 = 9x + 3$
   4. $2x - 5 = 3x - 5$
6. What must be true for the system of equations $y = mx + b$ and $y = nx + c$ to have infinite solutions?
   1. $m = n$ and $b \neq c$
   2. $m \neq n$
   3. $m = n$ and $b = c$
   4. $m \neq n$ and $b = c$
7. For the equation $a(x + 1) = ax + b$, what values of $a$ and $b$ will result in infinitely many solutions?
   1. $a = 1$, $b = 1$
   2. $a = 0$, $b = 0$
   3. $a = 5$, $b = 5$
   4. $a = 2$, $b = 3$